to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work

to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work

with Andrei Lerner and Sheldy Ombrosi.

with Andrei Lerner and Sheldy Ombrosi.

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===Stefanie Petermichl===

===Stefanie Petermichl===

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Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.

Abstract: Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>, and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.

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===Xianghong Chen===

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Title: Some transfer operators on the circle with trigonometric weights

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Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.

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===Bobby Wilson===

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Title: Projections in Banach Spaces and Harmonic Analysis

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Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.

Abstracts

Brian Street

Title: Convenient Coordinates

Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

Hiroyoshi Mitake

Title: Derivation of multi-layered interface system and its application

Abstract: In this talk, I will propose a multi-layered interface system which can
be formally derived by the singular limit of the weakly coupled system of
the Allen-Cahn equation. By using the level set approach, this system can be
written as a quasi-monotone degenerate parabolic system.
We give results of the well-posedness of viscosity solutions, and study the
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.

Joris Roos

Title: A polynomial Roth theorem on the real line

Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.

Michael Greenblatt

Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.

David Beltran

Title: Fefferman Stein Inequalities

Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.

Jonathan Hickman

Title: Factorising X^n.

Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.

Xiaochun Li

Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.

Fedor Nazarov

Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
conjecture is sharp.

Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
the norm of the Hilbert transform on the line as an operator from $L^1(w)$
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work
with Andrei Lerner and Sheldy Ombrosi.

Stefanie Petermichl

Title: Higher order Journé commutators

Abstract: We consider questions that stem from operator theory via Hankel and
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
more basic terms, let us consider a function on the unit circle in its
Fourier representation. Let P_+ denote the projection onto
non-negative and P_- onto negative frequencies. Let b denote
multiplication by the symbol function b. It is a classical theorem by
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
only if b is in an appropriate space of functions of bounded mean
oscillation. The necessity makes use of a classical factorisation
theorem of complex function theory on the disk. This type of question
can be reformulated in terms of commutators [b,H]=bH-Hb with the
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
as in the real variable setting, in the multi-parameter setting or
other, these classifications can be very difficult.

Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
spaces of bounded mean oscillation via L^p boundedness of commutators.
We present here an endpoint to this theory, bringing all such
characterisation results under one roof.

The tools used go deep into modern advances in dyadic harmonic
analysis, while preserving the Ansatz from classical operator theory.

Shaoming Guo

Title: Parsell-Vinogradov systems in higher dimensions

Abstract:
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.

Naser Talebizadeh Sardari

Title: Quadratic forms and the semiclassical eigenfunction hypothesis

Abstract: Let be any integral primitive positive definite quadratic form in variables, where , and discriminant . For any integer , we give an upper bound on the number of integral solutions of in terms of , , and . As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus for . This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.

Xianghong Chen

Title: Some transfer operators on the circle with trigonometric weights

Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.

Bobby Wilson

Title: Projections in Banach Spaces and Harmonic Analysis

Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.

Dong Dong

Title: Hibert transforms in a 3 by 3 matrix and applications in number theory

Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.